Light reception by a uniformly accelerated observer (${\tau }_{\mathrm{rec}}=-(c/a)\log (1-ad/c^2)$)

Let an observer start at rest at $T=0$, $X=R=c^2/a$ and accelerate towards increasing $X$ with constant proper acceleration $a>0$. A signal emitted simultaneously from $X=R-d$ travels along $X=R-d+cT$. The observer follows hyperbolic motion, $X=R\cosh (a\tau /c)$ and $cT=R\sinh (a\tau /c)$. Their intersection obeysA finite future reception exists only for $0\le d<c^2/a$. At equality the ray lies on the limiting null horizon and never reaches the observer; for larger separation it remains outside the observer's receivable past. The proper reception time tends to $d/c$ as $a\to 0$. The simultaneous emission and initial-rest conditions are essential; changing either changes the timing problem.